The normalized probability function of the Wilcoxon rank sum statistic is shown to converge to the normal density under very mild conditions on the two sample distributions. This is done by studying the conditional distribution of the rank sum given the first sample and by a rather heavy use of characteristic functions.
"A Local Limit Theorem for the Wilcoxon Rank Sum." Ann. Probab. 5 (6) 926 - 939, December, 1977. https://doi.org/10.1214/aop/1176995661